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Below is a list of documents that contain the definitive and tested equations (PDF) for the principal projections.
Note that each document has two parts: (1) Equations that calculate zone parameters that are only calculated once up front when projecting a group of points, and (2) Equations that are repeated for each point to be projected. This is done to prevent unecessary duplication in the calculations.
Provided are direct and inverse solutions.
Below are other documents that may be of interest in geomatics: